Extended explanation of Orevkov's paper on proper holomorphic embeddings of complements of Cantor sets in $\Bbb C^2$ and a discussion of their measure

TL;DR

This paper provides a detailed explanation of Orevkov's construction of proper holomorphic embeddings of complements of Cantor sets in c2b2, demonstrating that such embeddings can have Cantor sets with zero Hausdorff dimension.

Contribution

It offers an extended, clearer exposition of Orevkov's method for embedding c2b1b1c2b1b1c2b1b1c2b1b1c2b1b1c2b1b1c2b1b1c2b1b1 in c2b2 with Cantor sets of zero Hausdorff dimension.

Findings

Construction of proper holomorphic embeddings with Cantor sets of zero Hausdorff dimension.

Clarification of Orevkov's original cryptic construction.

Demonstration that such embeddings are possible with measure-zero Cantor sets.

Abstract

We clarify the details of a cryptical paper by Orevkov in which a construction of a proper holomorphic embedding $\gamma\colon\Bbb P^1\setminus{C}\hookrightarrow\Bbb C^2$ is performed; in particular, it is proved that such a construction can be done to get the Cantor set $C$ to have zero Hausdorff dimension.